Movement of anchorage teeth
The results of 46 teeth in terms of force component distribution, displacement and rotation movements in three directions were shown in Table 2. Regarding the force distribution of 46 teeth, very similar results were found for the 0.017×0.025-inch and 0.019×0.025-inch wires. The force components of the X-axis, Y-axis and Z-axis were approximately 1.36 N, 0.31 N and 0.45 N, respectively. Compared with the rectangular wire and round wire groups, the ribbon wire group had the lowest force component in three directions, and the force of the 0.025×0.017-inch group in the X-axis, Y-axis and Z-axis was 1.31 N, 0.29 N, 0.43 N, respectively. The round wire group displayed the highest X-axis force and Z-axis force (1.39 N and 0.48 N, respectively).
For the rectangular wire group, with increasing wire size, there was a descending trend of the displacement in the Y-axis and Z-axis. The displacement of the 0.017×0.025-inch wire in the Z-axis direction was 9.17 μm and that of the 0.019×0.025-inch wire was 9.13 μm. Among the three groups, ribbon wire (0.025×0.017-inch and 0.025×0.019-inch) exhibited the lowest displacement in the X-axis (12.61 μm and 12.77 μm, respectively) and Z-axis (8.99 μm and 9.06 μm, respectively). However, the 0.025×0.017-inch ribbon wire showed the highest Y-axis displacement. The Φ 0.018-inch round wire also showed higher Z-axis displacement (9.44 μm) and Y-axis displacement (5.96 μm) than Φ 0.020-inch round wire (9.20 μm and 4.40 μm, respectively).
In the rectangular wire group and ribbon wire group, the rotation motion for 46 teeth in the sagittal, frontal and occlusal planes was -0.002°, 0.020°, and 0.021°, respectively. In the ribbon wire group, the frontal rotation was much higher than that in the rectangular and round wire groups, especially the 0.025×0.017-inch group, which exhibited 0.042° of rotation. In the round wire group, Φ 0.020-inch wire displayed less rotation than Φ 0.018-inch wire, where the sagittal, frontal and occlusal rotation of Φ 0.020-inch wire was almost half of that of Φ 0.018-inch wire.
Von-Mises stress distribution of the mandibular PDL
Figure 2 illustrates the buccal view and occlusal view of the PDL Von-Mises stress distribution. In the posterior zone, the PDL of the first molar endured the most stress, especially in the distal root, and stress decreased gradually in the mesial direction. In the anterior zone, there was an increasing trend of stress from the canine tooth towards the central incisor. As shown in Figure 2, there was a very slight difference caused by the cross-sectional area in the PDL stress distribution, and the maximum stress was approximately 0.01 MPa.
Deformation of mandible dentition
The detailed movement (X-axis, Y-axis and Z-axis) of mandible teeth after the application of 1.5 N retraction force is shown in Figure 3. In the X-axis direction, all the teeth were inclined mesially due to the retracting force. As the cross-sectional area enlarged, the deformation of the X-axis was reduced in all groups. In the Y-axis direction, the posterior teeth were tipped to the lingual side, whereas the anterior teeth exhibited a labial-oriented motion. In the Z-axis direction, the first molar moved in an upward direction, but the central incisor had an opposite trend, indicating that the anterior teeth might intrude vertically since the posterior teeth tilted inward. The largest vertical deformation among mandible dentition was exhibited in the mandibular first molar, where the ribbon wire group exhibited the least vertical movement and the round wire group showed the highest deformation.
Deformation of the buccal line and lingual line
The displacements of the buccal line and lingual line are shown in Figure 4 (red refers to the buccal line, and blue refers to the lingual line). From the observation of the curve contour, the tendency of the buccal line and lingual line in different arch wire groups showed some similarity. The displacement was dramatically increased in the first molar region, followed by a slight decrease in the premolar zone before subsequently rising in the anterior teeth region.
In the rectangular wire group (0.017×0.025-inch and 0.019×0.025-inch), the maximum displacements of the buccal line and lingual line were approximately 0.016 mm and 0.014 mm, respectively. The movement of the first molar region was intermediate between the ribbon arch group and the round wire group. In the ribbon arch group, the lingual line deformation in the first molar region was approximately 0.008 mm, and it was also the lowest among the corresponding results of all the studied groups. Notably, the values of the 0.025×0.017-inch arch wire displacement, which were much higher than those of any other group, peaked at 0.019 mm in the central incisor region with a spike-like shape. In the 0.025×0.019-inch group, the displacements ranged from the first molar to the central incisor of the buccal line, and the lingual line was lowest among the studied groups. In the round wire group, the displacement ranges from the first molar to the central incisor of the buccal line and lingual line were higher than those of the rectangular and ribbon arch groups. The movement induced by the Φ 0.020-inch wire was approximately 0.0165 mm, which was lower than that induced by the Φ 0.018-inch wire.
Deformation of arch wire
Arch wire deformation is summarized in Figure 5. The rectangular wire group and ribbon wire group displayed a relative stress concentration in the anterior zone. The deformation range of the Φ 0.018-inch wire group was the largest in this study. The deformation range was almost the same in the rectangular wire group with different sizes, and the deformation value was approximately 0.007 mm.